The washer that sits on the inside face of the forward gear determines the end float play and is called a thrust washer. A collar on the propeller shaft pushes up against this thrust washer. Some may call it a "shim", but they are the same thing.
In the service manual it suggests an end float play between .005" - .020". In the manual it states to remove the shift yoke before measuring for end float. So the shift rod is not engaged with the prop shaft at all. But in order to do this you will have to remove the gear box cover for access to the shift yoke. Which isnt difficult but can be a pain to deal with as the shift yoke can easily fall off the end of the shaft when installing. Having it in gear may give you inaccurate end float measurements but dont quote me on that. I am just relaying the info in the manual.
Since the book calls for between .005" - .020" I think you may have to address it. Seems like .110" is excessive. Make sure the thrust washer is still on the forward gear and is not severely worn or the face of the forward gear it rides on is not worn.
The only "shim" BEHIND the forward gear is going to be behind the race for the forward gear roller bearing. Some have a shim there and some do not. Since you are not replacing the forward gear and its race I would not worry about this.
Double check that forward gear thrust washer and make sure its not damaged or missing. With that much play i bet the washer isnt even on there. You can find them on marineenginecom and look up your engine model number. They have varying thicknesses you can order. DO NOT put any ordinary washer in there. You will damage the gearbox in doing so. These shims ares specifically designed for this application.
The factory .059" thrust washer is on there, my dial calipers says it's only .057" but close enough. No apparent wear to the face of the forward gear where this washer resides or to the washer itself. What I find really weird, or at least I don't understand, is when I put the propeller back on there doesn't seem to be any shaft end float at all.